Homology, Homotopy and Applications

Volume 25 (2023)

Number 2

The cohomology of free loop spaces of rank $2$ flag manifolds

Pages: 343 – 372

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n2.a15


Matthew Burfitt (Yanqi Lake Beijing Institute of Mathematical Sciences and application (BIMSA), Beijing, China)

Jelena Grbić (School of Mathematics, University of Southampton, United Kingdom)


By applying Gröbner basis theory to spectral sequences algebras, we develop a new computational methodology applicable to any Leray–Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra. We demonstrate the procedure by deducing the cohomology of the free loop space of flag manifolds, presenting a significant extension over previous knowledge of the topology of free loop spaces. A complete flag manifold is the quotient of a Lie group by its maximal torus. The rank of a flag manifold is the dimension of the maximal torus of the Lie group. The rank $2$ complete flag manifolds are $SU(3)/T^2$, $Sp(2)/T^2$, $\mathit{Spin}(4)/T^2$, $\mathit{Spin}(5)/T^2$ and $G_2/T^2$. In this paper we calculate the cohomology of the free loop space of the rank $2$ complete flag manifolds.


cohomology, free loop space, flag manifold, spectral sequence, homotopy theory, Gröbner basis

2010 Mathematics Subject Classification

13P10, 14M15, 55P35, 55T10

Copyright © 2023, Matthew Burfitt and Jelena Grbić. Permission to copy for private use granted.

This research was supported in part by The Leverhulme Trust Research Project Grant RPG-2012-560.

Received 6 March 2022

Received revised 19 September 2022

Accepted 4 November 2022

Published 22 November 2023